Arithmetic Geometry & Local Fields

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Abstractions about arithmetic geometry and number theory — local-field theory (local field, higher local field, local class field theory, norm group), p-adic and crystalline cohomology (F-crystal, p-adic Hodge theory, Hodge-Arakelov theory), and arithmetic objects tied to curves and representations (arithmetic surface, Brandt matrix, Néron-Tate height).

14 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Arithmetic surface — A regular or suitably controlled two-dimensional scheme fibered over the spectrum of a Dedekind domain, with generic fiber an algebraic curve.
• Biquadratic field — A degree-four Galois extension of the rational numbers with Klein-four Galois group.
• Brandt matrix — A matrix encoding counts or weighted correspondences among ideal classes of a definite quaternion algebra, realizing Hecke operators on quaternionic modular forms.
• Deligne–Lusztig theory — A geometric construction of representations of finite groups of Lie type from compactly supported l-adic cohomology of varieties associated with reductive groups, Frobenius maps, and maximal tori.
• F-crystal — A finite free Witt-vector module equipped with an injective Frobenius-semilinear endomorphism, encoding crystalline-cohomological Frobenius structure.
• Geometric Langlands correspondence — A conjectural categorical correspondence relating local systems for a reductive group on an algebraic curve to sheaf-theoretic objects on the moduli stack of bundles for its Langlands dual group.
• Heegner's lemma — A descent lemma stating that a quartic curve with nonsquare leading coefficient has a rational point if it has a point over an odd-degree extension.
• Higher local field — A field equipped with an iterated tower of complete discrete valuations whose final residue field is finite or otherwise specified at dimension zero.
• Hodge–Arakelov theory — A proposed Arakelov-geometric analogue of Hodge comparison theory for elliptic curves, centered on functions on universal extensions and torsion points.
• Local class field theory — The theory that classifies finite abelian extensions of a local field through a reciprocity map from the field's multiplicative group to its abelian Galois group.
• Local field — A nondiscrete locally compact Hausdorff topological field, equivalently in the non-Archimedean case a complete discretely valued field with finite residue field, serving as a completion-scale model of global arithmetic.
• Norm group — The subgroup of a local field’s multiplicative group consisting of field norms from a finite abelian extension.
• Néron–Tate height — A canonical quadratic height on rational points of an abelian variety over a global field.
• P-adic Hodge theory — A comparison theory classifying p-adic Galois representations through period rings and their relations to de Rham, crystalline, semistable and Hodge–Tate cohomology.